The line-bundles tag has no wiki summary.
2
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1answer
97 views
When is the Clifford index of a curve computed by pencils?
Under which circumstances is the Clifford index of a curve computed by pencils?
-1
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2answers
158 views
Restriction of a line bundle to a two-cycle
I am reading a paper on Chiral Differential Operators
http://arxiv.org/pdf/hep-th/0604179v3.pdf
and it says on page 23 that a line bundle over a manifold C can be characterized completely by its ...
0
votes
1answer
171 views
Birkhoff decomposition vanishing of the Chern numbers
Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...
1
vote
1answer
88 views
Unicity of a vector field on $S^1$-bundle
Let M be a complex smooth manifold,and let $\zeta $ be a vector filed on $M$, why always there exists a unique vector field $\hat{\zeta }$ on $L^{\times}$ which project down to $\zeta $ and $\alpha( ...
-1
votes
4answers
282 views
An isomorphism on space of smooth sections
Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to
$$A=\{f:L^{\times}\to ...
1
vote
2answers
267 views
Isomorphism of connections on a complex line bundle
Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact?
Theorem. Let $E ...
1
vote
2answers
465 views
Uniqueness on square root of complex Line Bundle
Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?
1
vote
0answers
115 views
Hopf lemma for line bundles on curves in algebraic geometry
In the paper http://arxiv.org/pdf/math/0110256v1.pdf Claire Voisin proves that all linear subspaces which lie inside of a (not too big) secant variety of a smooth projective curve must lie inside one ...
0
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0answers
112 views
canonical bundle of the relative spectrum
maybe it is a very trivial quetion but:
suppose we have a smooth projective variety $X$ over $k$ and $\mathcal{A}$ an $\mathcal{O}_X$ algebra. We have the relative spectrum ...
0
votes
1answer
170 views
Could we construct the Jacobian variety of a smooth curve $C$ with genus $>2$ from its derived category $D(C)$?
Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of ...
4
votes
0answers
223 views
Ample Line Bundles on Algebraic Spaces
The sources known to me (Knutson's Algebraic Spaces and Pascual-Gainza's Ampleness criteria for algebraic spaces) define a line bundle $L$ on an algebraic space $X$ (over a base scheme $S$) to be ...
0
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0answers
51 views
on relative divisors over artinian rings
Let $X$ a curve over $\mathbb{C}$, $D$ a divisor on $X$, $R$ a local artinian ring of residue field $\mathbb{C}$
Let $A=H^{0}(X_{R},\mathcal{O}(D_{R}))$ the scheme of sections over $Spec(R)$.
Let ...
0
votes
1answer
284 views
A reference about Dolbeault cohomology
I am looking for a reference about Dolbeault cohomology when the line bundle is not supposed to be positive.
1
vote
1answer
203 views
extension of torsion line bundles
Assume that you are given a smooth projective algebraic variety $X$ and a divisor $D$ with normal crossings on it. Put $U=X-D$. Assume now that $L$ is a torsion line bundle(=invertible sheaf) on $U$, ...
7
votes
1answer
240 views
Vanishing theorems in positive characteristic
In the paper
Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078
I found the ...
10
votes
2answers
1k views
What is Kirillov's method good for?
I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...
0
votes
1answer
107 views
Trivial Line Bundle-Riemann surfaces
What are the Hermitian metrics in a trivial line bundle on a Riemann surface X?
I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ weight function ...
0
votes
0answers
165 views
canonical model of a reducible curve
Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) ...
3
votes
2answers
273 views
Line bundles on K3 surfaces
Let $L$ be a line bundle on an (algebraic) K3 surface over a field $k$. The Riemann-Roch theorem specializes to
$$
\chi(X, L)=\frac{1}{2}(L\cdot L)+2
$$
which can be rewritten as
$$
h^0(X, ...
7
votes
2answers
496 views
How many flat connections has a line bundle in algebraic geometry?
Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how ...
2
votes
1answer
311 views
On morphisms to projective space arising from a linear system
Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...
1
vote
1answer
186 views
Extending line bundles
Suppose you have a one parameter family of algebraic varieties over unit disk, such that the central fiber is singular and is a union (normal-crossing) of two varieties and the rest are smooth.
Is it ...
2
votes
1answer
170 views
divisors and powers of line bundles
Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ and $m \geq 2$ an ...
1
vote
0answers
189 views
How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
How does the line bundles look like on a proper model (or Néron model) of an abelian variety?
Who knows references about this?
In particular, let us work over a trait $S=\mathrm{Spec} R$, where $R$ ...
7
votes
1answer
459 views
Non-compact Kähler manifolds which admit a positive line bundle
A complex manifold which admits a positive line bundle is automatically Kähler. Furthermore, if the manifold is compact, then it is projective by the Kodaira Embedding Theorem. In particular, not ...
1
vote
0answers
101 views
pairing theta functions for different complex structures
I apologize for my previous attempt to ask this, which was very badly written.
Let us start with $\mathbb{C}\times\mathbb{C}$. To form an Hermitian line bundle over a complex torus with complex ...
0
votes
0answers
243 views
Theta functions and Fourier transforms
Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal ...
6
votes
4answers
767 views
Cohomology of line bundles
For sure answers to my questions are well known - but I never saw them anywhere.
Let $X$ be a smooth projective (or just proper) variety over an algebraically closed field $k$. Let $A_i$ be the ...
7
votes
3answers
390 views
line bundles and universal covers
When dealing with some lifting problems, I came across the following problem, which probably has a well-known answer, but anyway:
Suppose I have a (locally contractible) topological group $G$, such ...
2
votes
3answers
328 views
Pedagogical notes on line bundles on complex projective manifolds
I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems ...
9
votes
1answer
539 views
Is there a mathematical explanation for the Aharonov-Casher effect?
Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows.
Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential ...
1
vote
1answer
180 views
etale covers of line bundles on an abelian variety
subj: etale covers of line bundles on an abelian variety
Is there an explicit decryption of finite
etale covers of a line bundle $L$ on an abelian variety and its associated C*-bundles
$L^o = L ...
3
votes
1answer
561 views
Connections on line bundles over the torus
If I understand correctly, every line bundle $L$ over the (2-dim) torus can be obtained from a quotient of $\mathbb{R}^2 \times \mathbb{C}$ by a $\mathbb{Z}^2$ lattice action. Different line bundles ...
0
votes
2answers
500 views
What is the geometric point of view of an algebraic line bundle compared to a analytic line bundle?
Hi folkz,
I'm trying to learn more about line bundles, invertible sheaves and divisors on schemes. I understand the connection beweteen Cartier and Weil Divisors and the connection between Cartier ...
2
votes
1answer
441 views
What can be said about a pullback of a very ample line bundle w.r.t birational maps?
Let $X$ be a smooth projective variety and $\phi: X \to \mathbb P^n$ be a map. If $\phi$ is an embedding then $E=\phi^*(O(1))$ is very ample. But can one say something if $\phi$
is birational (but not ...
1
vote
2answers
906 views
Line bundles with complex connection
Suppose that we have a complex manifold $X$, and a line bundle $L$ over $X$. It is known that the line bundles over $X$ are parametrized by their Chern class, the Chern class being the cohomology ...
4
votes
2answers
692 views
Is the double-twisted Moebius strip isotopic to the trivial strip?
Abstractly, on the topological circle $S^1$ there are only two real line bundles, up to isomorphism: the trivial one $\mathcal{O}$ and the Moebius strip $\mathcal{O}(1)$ (thinking of $S^1$ as ...
3
votes
2answers
203 views
Uniformity of injectivity for maps associated to linear systems
Let $X$ be a compact complex manifold and $L\to X$ a holomorphic line bundle (without any a priori assumption on its positivity).
Suppose that for each $x,y\in X$, with $x\ne y$, there exists a ...
3
votes
0answers
192 views
How can one check that two line bundles on $\overline{M}_{0,n}$ coincide?
Let $X$ be the Deligne-Mumford compactification of $\mathcal{M}_{0,n}$. Suppose I have two (big) line bundles $L$ and $L'$ on $X$ and that I want to show that they are the same element of $Pic(X)$. Of ...
3
votes
1answer
297 views
Line bundles on Ind Schemes
I've learned when you have a integral smooth scheme line bundles are the same as Cartier divisors are the same as Weil divisors. My question is to what extent does this continue to hold (if at all) ...
1
vote
2answers
289 views
Connections with compatible Hermitian products on complex line bundles
Let $X$ be a manifold, $L$ be a complex line bundle over $X$, and $L^{*}$ be the associated principal bundle. Suppose $\alpha$ is a connection form on $L^{*}$, with associated connection $D$ on $L$. ...
0
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0answers
167 views
Does the normalization of a projective morphism determine the line bundle?
Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms
$$f:X \to \mathbb{P}^n$$
and
$$g:X \to \mathbb{P}^m,$$
such that the image of $f$ is the ...
1
vote
1answer
209 views
Does the semi-stable set determine the linearization of a GIT quotient?
Suppose I have a morphism $f:X\to Y$ which is a GIT quotient of $X$ with respect to some reductive, linear group.
Does the semistable $X^{ss}$ and stable locus $X^s\subset X$ determine completely the ...
2
votes
1answer
443 views
Line bundles, linear systems and normalization
One example that I always have in mind is that the plane nodal (or even the plane cuspidal) cubic curve $X$ is obtained by an appropirate 2-dim linear subsystem of $|\mathcal{O} (3)|$ on ...
5
votes
0answers
225 views
Do simplicial toric varieties have “lots” of base point free linear systems?
Question: Let $n$ be a positive integer and let $X$ be a simplicial toric variety. Does every coset of $n\cdot Pic(X)\subseteq Pic(X)$ contain a base point free linear system?
If $X$ is not ...
5
votes
3answers
451 views
Line bundles on fibrations
Let $f:Y \to X$ be a flat morphism with positive dimensional fibers. Is it always true that line bundles that are trivial along each fiber are of type $f^*L$ for $L$ a line bundle on $X$?
9
votes
1answer
575 views
Are bundle gerbes bundles of algebras?
The category of line bundles (possibly with connection)
on a smooth manifold M can be defined in two different ways:
The first definition uses transition functions that satisfy a cocycle condition
...
4
votes
1answer
896 views
Cohomology theory for symplectic manifolds
Suppose I have a symplectic manifold $(M,\omega)$ and a line bundle $\mathcal L$ with a connection with curvature $\omega$ (or perhaps it's more standard to say $\frac i{2\pi}\omega$; anyway, the ...
3
votes
1answer
435 views
Pulling back a line bundle on the Jacobian to a spin bundle on the curve
I'd like to have an expression for the (or some) line bundle on the Jacobian $J$ of a smooth complex projective curve $C$ with genus $g >1$ which pulls back to a chosen spin bundle (theta ...
1
vote
1answer
167 views
Followup; Strict Transform of a Line Bundle
This is a follow up to my previous question, and I have lowered my demands to a situation as follows:
Let $X$ be an algebraic variety, $\mathcal{I}$ a coherent sheaf of ideals and $\mathcal{L}$ a ...
